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Exam TFY4205 - Quantum Mechanics II

Academic contact during examination: Jacob Linder Phone: 951 73 515

Examination date: December 11th, 2019 Examination time (from-to): 15:00-19:00

Permitted examination support material: Karl Rottman, Mathematische Formelsammlung and a simple calculator.

Read each problem carefully. The maximum number of points each problem can give (perfect score) is shown in the title of each problem. Good luck.

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Problem 1 (1 point)

The following commutation relation is correct for fermionic creation and annihilation operators:

Velg ett alternativ

Maks poeng: 1

2

Problem 2 (1 point)

If particles with momentum approach a potential with range , a semiclassical analysis would tell us that only particles with angular momentum quantum number satisfying the following equation would be scattered.

Maks poeng: 1

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2/10 Assume is an exactly solvable Hamilton-operator, meaning all quantities in are

known. If we add a small perturbation to the Hamilton-operator, the eigenstates of are denoted In the presence of the perturbation to the Hamilton-operator, the lowest order correction to the unperturbed energy eigenvalues takes the form

Maks poeng: 1

4

Problem 4 (1 point)

In the variational method, the key idea is that:

The ground-state energy is an upper limit for the energy expectation value of any trial wavefunction Choosing a trial wavefunction and minimizing it with respect to its variables, yields the correct ground- state energy of the system

The wavefunction varies so slowly that one may write the wavefunction as a plane-wave with a wavenumber weakly dependent on position

The expectation value of the Hamilton-operator in any state must be greater or equal to the ground-state energy

Maks poeng: 1

5

Problem 5 (1 point)

The WKB-approximation is expected to work well when:

The energy of the particle varies equally fast as the potential The energy of the particle is small compared to its rest mass The potential in the system varies rapidly in space

The potential in the system varies slowly in space

Maks poeng: 1

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Problem 6 (1 point)

The concept of detailed balance in time-dependent perturbation theory states that:

To first order in the perturbation, the probability for a transition between two states is equal to the probability for the opposite transition

To first order in the perturbation, a harmonically varying potential will give rise to resonances

To first order in the perturbation, the relative probabilities of a transition between two states and the opposite transition depend on the details of the potential

To first order in the perturbation, the frequency of the potential matches the energy level spacing

Maks poeng: 1

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4/10 NB! In the following problem, you can derive the correct criterion from the information given below - this is not

about simply memorizing the correct answer. You may find it useful to recall the completeness relation

Consider now the following problem.

In the method of sudden approximation, one uses continuity of the wavefunction along the time-axis to derive a criterion for when the sudden approximation is supposed to work well.

Imagine that the eigenstates and eigenvalues for two time-independent Hamilton-operators are known. A general solution of the time-dependent Schrödinger equation is a superposition of stationary states.

Imagine that the system changes abruptly from at t=0. It is then possible to compute an expression for the probability coefficients of the state of the system in the time interval when the system is described by

. Let this procedure be known as the "first approach".

Now we refine this model a bit. Let the Hamilton-operator be

The corresponding eigenvalues of these operators, which individually are time-independent, may be denoted The eigenstates for all three Hamilton-operators are also known. It is then again possible to compute an expression for the probability coefficients of the state of the system in the time interval where the system is described by . Let this

procedure be known as the "second approach".

The correct criterion for when the first approach will be a good approximation for the result obtained in the second approach is:

Maks poeng: 3

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Problem 8 (1 point)

The adiabatic theorem in quantum mechanics states that:

An adiabatically varying Hamiltonian will give rise to a non-zero dynamical phase so long as the Hamilton-operator is not cyclic

A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the energy spectrum

The transition between two instantaneous eigenstates caused by a perturbation that varies adiabatically vanishes if there is no gap between the energy levels of those states

The forward-scattering amplitude determines the total scattering cross section

Maks poeng: 1

9

Problem 9 (1 point)

The Berry phase in quantum mechanics is:

A phase picked up by the wavefunction of a cyclic Hamilton-operator as one cycle is completed An example of how the phase of the wavefunction at a given time is physically observable

Similar to the dynamical phase of the system in that it depends on how fast the Hamiltonian changes The phase at any instant of time for a cyclic Hamilton-operator

Maks poeng: 1

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6/10

Consider a normalized Schrödinger equation written as .

Let the Green function be defined by the equation .

Let be the solution to the Schrödinger equation with

If is known, then the Schrödinger equation is equivalent to the integral equation:

Maks poeng: 1

11

Problem 11 (1 point)

In the Born-approximation, the scattered wavefunction is given by:

.

When a spherically-symmetric scattering potential satisfies the property , then the following statement is true.

The Born-approximation is expected to give good results for all energies

This property is irrelevant with regard to the validity of the Born-approximation, since the energy is not part of the criterion

The Born-approximation is not expected to work well for all energies

The Born-approximation is expected to give good results for low energies

Maks poeng: 1

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Problem 12 (1 point)

In the method of partial waves (in the context of scattering), the scattering amplitude is given by

In this expression, the quantity is:

Obtained from the phase of the scattering potential at large distances

Obtained from a semiclassical argument for which incident waves that are scattered Obtained from the phase of the radial part of the wavefunction at large distances Obtained from the phase of the angular part of the wavefunction

Maks poeng: 1

13

Problem 13 (3 points)

When deriving the optical theorem from particle flux conservation, one arrives at the following equation as an intermediate step:

where r is assumed very large ( ).

You may find the following information useful:

Solve the integral and identify the correct expression for the total scattering cross section.

Maks poeng: 3

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8/10 Two identical fermions with in a spin-triplet state scatter on each other. The correct spatial asymptotic

wavefunction is then proportional to:

where r is the relative coordinate for the fermions. The meaning of the symbols should be known by the student.

Maks poeng: 1

15

Problem 15 (1 point)

Consider the time-dependent Schrödinger equation

where is a potential energy term independent on the charge of the wavefunction . Gauge-invariance for this equation then means that:

One can replace the wavefunction with and obtain the same physical situation as with .

One can replace the vector potential and obtain the same physical situation as when using .

It does not matter which phase the vector potential has, since its absolute value is independent on the phase.

One can replace with and and obtain the same

physical situation as with .

Maks poeng: 1

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Problem 17 (1 point)

Which of the following statements regarding is incorrect:

At zero temperature, electrons moving through a perfect crystal lattice will experience a finite resistance due to phonon scattering

Phonons will cause electrons to scatter when moving through a crystal structure Electron states in crystals may be described by energy bands which may have gaps

If a crystal structure in which electrons move have defects, this will cause electrical resistance

Maks poeng: 1

17

Problem 18 (4 points)

A particle with mass m moves in 1D in a periodic potential which has an infinite extent. The potential has the form

where are positive constants.

(a) Derive appropriate boundary conditions for the wavefunction and its derivative to be used in the immediate vicinity of the positions .

(b) Let the lowest energy of a wave that can propagate through this potential be defined as Use the boundary conditions from (a) and the property

for a wavefunction in a periodic potential, where is a real constant, to derive a transcendental equation (not a differential equation) that can be solved to give . It is not necessary to solve the transcendental equation you find in this manner.

Maks poeng: 4

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10/10 Consider s-wave scattering (meaning angular momentum quantum number ) of a particle moving in the

potential . Here, .

(a) Show that the wavefunction

satisfies the Schrödinger equation. Derive the relation between k and k' which both are assumed positive and real. You may find it useful to know that the Laplace operator in spherical coordinates acting on a function f takes the form

(b) Let the logarithmic derivative of the wavefunction be defined as Explain what the appropriate boundary condition for the logarithmic derivation is at

(c) Use the boundary condition from (b) to derive an explicit analytical expression for the phase-shift

Maks poeng: 3

19

Problem 19 (4 points)

The quantum mechanical operator for the electric field may be written:

where i in front of the summation is the imaginary number and is a unit length polarization vector.

A coherent photon state for a given mode is expressed as

where we have dropped the mode indices for simplicity. The coherent state is normalized to unity and the coefficients satisfy . The operators satisfy

and where again mode indices were omitted for brevity of notation.

Use the above information derive an analytical expression for the standard deviation

when the system is in a coherent photon state for a given mode .

Maks poeng: 4